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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arymaptfv | Structured version Visualization version GIF version |
Description: The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.) |
Ref | Expression |
---|---|
1arymaptfv.h | ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩}))) |
Ref | Expression |
---|---|
1arymaptfv | ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6884 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{⟨0, 𝑥⟩}) = (𝐹‘{⟨0, 𝑥⟩})) | |
2 | 1 | mpteq2dv 5243 | . 2 ⊢ (ℎ = 𝐹 → (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩}))) |
3 | 1arymaptfv.h | . 2 ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩}))) | |
4 | eqid 2726 | . . . . 5 ⊢ (0..^1) = (0..^1) | |
5 | 4 | naryrcl 47597 | . . . 4 ⊢ (ℎ ∈ (1-aryF 𝑋) → (1 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
6 | 5 | simprd 495 | . . 3 ⊢ (ℎ ∈ (1-aryF 𝑋) → 𝑋 ∈ V) |
7 | 6 | mptexd 7221 | . 2 ⊢ (ℎ ∈ (1-aryF 𝑋) → (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})) ∈ V) |
8 | 2, 3, 7 | fvmpt3 6996 | 1 ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 ⟨cop 4629 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 ℕ0cn0 12476 ..^cfzo 13633 -aryF cnaryf 47592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-naryf 47593 |
This theorem is referenced by: 1arymaptf1 47608 |
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